arXiv subjects
Publications and source records attributed to Paolo Lisca.
We determine the closed, oriented Seifert fibered 3-manifolds which carry positive tight contact structures. Our main tool is a new non-vanishing criterion for the contact Ozsvath-Szabo invariant.
We classify connected sums of three-dimensional lens spaces which smoothly bound rational homology balls. We use this result to determine the order of each lens space in the group of rational homology 3-spheres up to rational homology cobordisms, and to determine the concordance order of each 2-bridge knot.
We apply Donaldson's theorem on the intersection forms of definite 4--manifolds to characterize the lens spaces which smoothly bound rational homology 4--dimensional balls. Our result implies, in particular, that every smoothly slice 2--bridge knot is ribbon, proving the ribbon conjecture for 2--bridge knots.
In this paper we prove a vanishing theorem for the contact Ozsvath--Szabo invariants of certain contact 3--manifolds having positive Giroux torsion. We use this result to establish similar vanishing results for contact structures with underlying 3--manifolds admitting either a torus fibration over the circle or a Seifert fibration over an orientable base. We also show -- using standard techniques from contact topology -- that if a contact 3--manifold (Y,ξ) has positive Giroux torsion then there exists a Stein cobordism from (Y,ξ) to a contact 3--manifold (Y,ξ') such that (Y,ξ) is obtained from (Y,ξ') by a Lutz modification.
Let C be the contact structure naturally induced on the lens space L(p,q) by the standard contact structure on the three--sphere. We obtain a complete classification of the symplectic fillings of (L(p,q),C) up to orientation-preserving diffeomorphisms. In view of our results, we formulate a conjecture on the diffeomorphism types of the smoothings of complex two-dimensional cyclic quotient singularities.
We classify tight contact structures on the small Seifert fibered 3--manifold M(-1; r_1, r_2, r_3) with r_i in (0,1) and r_1, r_2 \geq 1/2. The result is obtained by combining convex surface theory with computations of contact Ozsvath--Szabo invariants. We also show that some of the tight contact structures on the manifolds considered are nonfillable, justifying the use of Heegaard Floer theory.
We characterize L-spaces which are Seifert fibered over the 2-sphere in terms of taut foliations, transverse foliations and transverse contact structures. We give a sufficient condition for certain contact Seifert fibered 3-manifolds with e_0=-1 to have nonzero contact Ozsvath--Szabo invariants. This yields an algorithm for deciding whether a given small Seifert fibered L-space carries a contact structure with nonvanishing contact Ozsvath--Szabo invariant. As an application, we prove the existence of tight contact structures on some 3-manifolds obtained by integral surgery along a positive torus knot in the 3-sphere. Finally, we prove planarity of every contact structure on small Seifert fibered L-spaces with e_0 bigger than or equal to -1, and we discuss some consequences.
We prove various results on contact structures obtained by contact surgery on a single Legendrian knot in the standard contact three--sphere. Our main tool are the contact Ozsvath--Szabo invariants.
We characterize the oriented Seifert-fibered three-manifolds which admit positive, transverse contact structures.
Let p and n be positive integers with p>1, and let E(p,n) be the oriented 3-manifold obtained by performing pn(p-1)-1 surgery on a positive torus knot of type (p, pn+1). We prove that E(2,n) does not carry tight contact structures for any n, while E(p,n) carries tight contact structures for any n and any odd p. In particular, we exhibit the first infinite family of closed, oriented, irreducible 3-manifolds which do not support tight contact structures. We obtain the nonexistence results via standard methods of contact topology, and the existence results by using a quite delicate computation of contact Ozsvath-Szabo invariants.
Let S^3_r(K) be the oriented 3--manifold obtained by rational r-surgery on a knot K in S^3. Using the contact Ozsvath-Szabo invariants we prove, for a class of knots K containing all the algebraic knots, that S^3_r(K) carries positive, tight contact structures for every r not= 2g_s(K)-1, where g_s(K) is the slice genus of K. This implies, in particular, that the Brieskorn spheres -Sigma(2,3,4) and -Sigma(2,3,3) carry tight, positive contact structures. As an application of our main result we show that for each m in N there exists a Seifert fibered rational homology 3-sphere M_m carrying at least m pairwise non-isomorphic tight, nonfillable contact structures.
We classify positive, tight contact structures on closed Seifert fibered 3-manifolds with base S^2, three singular fibers and e_0\geq 0.
Using contact surgery we define families of contact structures on certain Seifert fibered three-manifolds. We prove that all these contact structures are tight using contact Ozsath-Szabo invariants. We use these examples to show that, given a natural number n, there exists a Seifert fibered three-manifold carrying at least n pairwise non-isomorphic tight, not fillable contact structures.
We prove that an infinite family of virtually overtwisted tight contact structures discovered by Honda on certain circle bundles over surfaces admit no symplectic semi-fillings. The argument uses results of Mrowka, Ozsvath and Yu on the translation-invariant solutions to the Seiberg-Witten equations on cylinders and the non-triviality of the Kronheimer-Mrowka monopole invariants of symplectic fillings.
Extending our earlier results, we prove that certain tight contact structures on circle bundles over surfaces are not symplectically semi--fillable, thus confirming a conjecture of Ko Honda.
The standard contact structure on the three-sphere is invariant under the action of the cyclic group of order p yielding the lens space L(p,q). Therefore, every lens space carries a natural quotient contact structure Q. A theorem of Eliashberg and McDuff classifies the symplectic fillings of (L(p,1), Q) up to diffeomorphism. We announce a generalization of that result to every lens space. In particular, we give an explicit handlebody decomposition of every symplectic filling of (L(p,q), Q) for every p and q. Our results imply that: (a) there exist infinitely many lens spaces L(p,q) with q>1 such that (L(p,q), Q) admits only one symplectic filling up to blowup and diffeomorphism; (b) for any natural number N, there exist infinitely many lens spaces L(p,q) such that (L(p,q), Q) admits more than N symplectic fillings up to blowup and diffeomorphism.
This paper has been withdrawn by the authors, due to a mistake pointed out by Lenny Ng and Josh Sabloff.
Let X be a 4-manifold with contact boundary. We prove that the monopole invariants of X introduced by Kronheimer and Mrowka vanish under the following assumptions: (i) a connected component of the boundary of X carries a metric with positive scalar curvature and (ii) either b_2^+(X)>0 or the boundary of X is disconnected. As an application we show that the Poincare homology 3-sphere, oriented as the boundary of the positive E_8 plumbing, does not carry symplectically semi-fillable contact structures. This proves, in particular, a conjecture of Gompf, and provides the first example of a 3-manifold which is not symplectically semi-fillable. Using work of Froyshov, we also prove a result constraining the topology of symplectic fillings of rational homology 3-spheres having positive scalar curvature metrics.